Konrad-Zuse-Zentrum für Informationstechnik Berlin Heilbronner Str. 10, D Berlin - Wilmersdorf
|
|
- Florence Miller
- 5 years ago
- Views:
Transcription
1 Konrad-Zuse-Zentru für Inforationstechnik Berlin Heilbronner Str. 10, D Berlin - Wilersdorf Folkar A. Borneann On the Convergence of Cascadic Iterations for Elliptic Probles SC 94-8 (Marz 1994)
2 1 0
3 ON THE CONVERGENCE OF CASCADIC ITERATIONS FOR ELLIPTIC PROBLEMS FOLKMAR A. BORNEMANN Abstract. We consider nested iterations, in which the ultigrid ethod is replaced by soe siple basic iteration procedure, and call the cascadic iterations. They were introduced by Deuhard, who used the conugate gradient ethod as basic iteration (CCG ethod). He deonstrated by nuerical experients that the CCG ethod works within a few iterations if the linear systes on coarser triangulations are solved accurately enough. Shaidurov subsequently proved ultigrid coplexity for the CCG ethod in the case of H 2 -regular two-diensional probles with quasi-unifor triangulations. We show that his result still holds true for a large class of soothing iterations as basic iteration procedure in the case of two- and three-diensional H 1+ -regular probles. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new terination criterion for the CCG ethod. Key Words. Finite eleent approxiation, cascadic iteration, nested iteration, soothing iteration, conugate gradient ethod, adaptive triangulations AMS(MOS) subect classication. 65F10, 65N30, 65N55 Introduction. Let R d be a polygonal Lipschitz doain. We consider an elliptic Dirichlet proble on in the weak forulation: u 2 H 1 0() : a(u; v) = (f; v) L 2 8v 2 H 1 0(): Here f 2 H?1 () and a(; ) is assued to be a H 1 0 ()-elliptic syetric bilinear for. The induced energy-nor will be denoted by kuk 2 a = a(u; u) 8u 2 H 1 0(): Given a nested faily of triangulations (T )` eleents are the spaces of linear nite X = fu 2 C() : u T 2 P 1 (T ) 8T 2 T ; = 0g; where P 1 (T ) denotes the linear functions on the triangle T. We have X 0 X 1 : : : X` H 1 0(): The nite eleent approxiations are given by u 2 X : a(u ; v ) = (f; v ) L 2 8v 2 X : Fachbereich Matheatik, Freie Universitat Berlin, Gerany, and Konrad-Zuse-Zentru Berlin, Heilbronner Str. 10, Berlin, Gerany. borneann@sc.zib-berlin.de 1
4 For ne eshes T` the direct coputation of u` is a prohibitive expensive coputational task, so one uses iterative ethods. With the choice of soe basic iterative procedure I, the following cascadic iteration akes use of the nested structure of the spaces X : (1) (i) u 0 = u 0 (ii) = 1; : : : ; ` : u = I ; u?1 : Here I ; denotes steps of the basic iteration applied on level. This kind of iteration is known in the literature under dierent naes, depending on the choice of the basic iteration and the paraeter : Nested iteration: the basic iteration is a ultigrid-cycle, the paraeters are chosen a priori as a sall constant nuber of iterations, cf. Hackbusch [5]. Cascade: the basic iteration is a ultilevel preconditioned cg-ethod, the are chosen a posteriori due to certain terination criteria. This ethod was naed and invented by Deuhard, Leinen and Yserentant [4]. CCG-iteration: the basic iteration is a plain cg-ethod, the are chosen a posteriori. CCG stands for cascadic conugate gradient ethod and was introduced by Deuhard [3]. We call a cascadic iteration optial for level `, if we obtain accuracy ku`? u `k a ku? u`k a ; i.e. if the iteration error is coparable to the approxiation error, and if we obtain ultigrid coplexity aount of work = O(n`); where n` = di X`. The optiality of the nested iteration and of Cascade are well known [5, 4], at least for certain situations. The optiality of the CCG ethod has been deonstrated by several nuerical exaples [3]. This has been considered as rather astonishing, since only a plain basic iteration is used. Shaidurov [10] has recently shown for H 2 -regular probles and quasiunifor triangulations in two diension, that the CCG ethod is optial for a certain choice of the paraeters. Since he shows in essence, that the cg-ethod has soe soothing properties, we were lead to consider rather general soothers as basic iterations. The ain result of Section 1 is as follows: For H 1+ -regular probles with 0 < 1 and quasi-unifor triangulations it is possible to choose the paraeters, that 2
5 for d = 3: the cascadic iteration is optial for level `. for d = 2: the cascadic iteration is accurate and has coplexity O(n` log n`). This result holds for a large class of soothing iterations. In Section 2 we show how Shaidurov's result ts in our frae. Finally in Section 3 we show for adaptive grids how to choose the a posteriori by certain terination criteria. Under soe heuristically otivated assuptions on the adaptive grids we are able to show optiality for the cg-ethod as basic iteration. Reark. With respect to the CCG ethod, we decided to call the algorith (1) cascadic iteration. Since the interesting case here is the use of plain basic iterations, we had to choose a nae dierent fro nested iteration. 1. General soothers as basic iteration. In this and the following section we consider quasi-unifor triangulations with eshsize paraeter 1 c 2? h = ax T 2T dia T c 2? : In the following we use the sybol c for any positive constant, which only depends on the bilinear for a(; ), on and the shape regularity as well as the quasi-unifority of the triangulations. All other dependencies will be explicitly indicated. A general assuption on the elliptic proble will be H 1+ -regularity for soe 0 < 1, i.e., kuk H 1+ c kfk H?1 8f 2 H?1 (): The approxiation error of the nite eleent ethod is then given in energy nor as (2) ku? u k a c h kfk H?1 = 0; : : : ; `; cf. [6, Lea 8.4.9]. By the well-known Aubin-Nitsche lea and an interpolation arguent one gets the approxiation property (3) ku? u?1 k H 1? c h ku? u?1 k a = 1; : : : ; `; cf. [6, Theore ]. We consider the following type of basic iterations for the nite-eleent proble on level started with u 0 2 X : u? I ; u 0 = S ; (u? u 0 ) 3
6 with a linear apping S ; : X! X for the error propagation. We call the basic iteration a soother, if it obeys the soothing property (4) (i) ks ; v k a c h?1 kv k L 2 8v 2 X ; (ii) ks ; v k a kv k a with a paraeter 0 < 1. As is shown in [5] the syetric Gau-Seidel-, the SSOR- and the daped Jacobi-iteration are soothers in the sense of (4) with paraeter = 1=2: Lea 1.1. A soother in the sense of (4) fullls ks ; v k a c h? kv k H 1? 8v 2 X : Proof. This can be shown be an usual interpolation arguent using discrete Sobolev nors like those introduced in [1] and their equivalence to the fractional Sobolev nors. We are now able to state and prove the ain convergence estiate for the cascadic iteration (1). Theore 1.2. The error of the cascadic iteration with a soother as basic iteration can be estiated by 1 ku`? u `k a c ku? u?1 k a c h kfk H?1: Proof. For = 1; : : : ; ` we get by the linearity of S ; ku? u k a = ks ; (u? u?1 )k a ks ; (u? u?1 )k a + ks ; (u?1? u?1)k a : The rst ter can be estiated by Lea 1.1 and the approxiation property (3): ks ; (u? u?1 )k a c h? 4 c 1 ku? u?1 k H 1? ku? u?1 k a :
7 If we estiate the second ter by property (4(ii)) of a soother, we get ku? u k a c Using u 0 = u 0 we get by induction ku`? u `ka c Galerkin orthogonality gives ku? u?1 k a + ku?1? u?1k a : 1 ku? u?1 k a : ku? u?1 k a ku? u?1 k a ; so that the error estiate (2) yields the second assertion. We choose as the sallest integer, such that (5) 2 (d+1)(`?)=2 : The integer = ` is therefore the nuber of iterations on level `. With this choice the cascadic iteration can be shown to be optial. Lea 1.3. Let the nuber of iterations on level be given by (5). The cascadic iteration yields the error if > 1=d, and if = 1=d. ku`? u `k a c() h` kfk H?1; ku`? u `k a c Proof. By 2? =c h c 2? we get h h` (1 + log h`) =d kfk H?1 c? 2?(d+1)`=2 2 (d?1)=2 : If > 1=d this is a geoetric su which can be estiated by h c()? 2?(d+1)`=2 2 (d?1)`=2 = c()? 2?` c()? h ` : In the case = 1=d the su is equal to `, such that h =d c?=d 2?`` c?=d h ` (1 + log h`): 5
8 Theore 1.2 yields the assertion. The coplexity of the ethod is given by the following Lea 1.4. Let the nuber of iterations on level be given by (5). If > 1=d we get n c() n`; if = 1=d n c n`(1 + log n`): Proof. We have 2 d =c n = di X c 2 d : Therefore we get n c 2 (d+1)`=2 2 (d?1)=2 : If > 1=d this is a geoetric su which can be estiated by n c() 2 (d+1)`=2 2 (d?1)`=2 = c()2 d` c() n`: In the case = 1=d the su is equal to `, such that n c 2 d` ` c n`(1 + log n`): In order that also in the case = 1=d the cascadic iteration has an iteration error like the discretization error (2), we choose a special nuber of nal iterations. Lea 1.5. Let = 1=d. If we choose the nuber of iterations on level ` as the sallest integer with (1 + log h`) d= we get for the error of the cascadic iteration ku`? u `k a c 6 h` =d kfk H?1;
9 and as coplexity n c n`(1 + log n`): Proof. By observing that c 0 ` (1 + log h`) c 1 (1 + log n`) c 2 ` the assertion is clear by Lea 1.3 and Lea 1.4. Our results show, that the cascadic iteration with a plain Gauss-Seidel-, SSOR- or daped Jacobi-iteration (all with = 1=2) as basic iteration is optial for d = 3, accurate with coplexity O(n` log n`) for d = Conugate gradient ethod as basic iteration. When using the conugate gradient ethod as the basic iteration we have to tackle with a proble: the result I ; u 0 of steps of the cg-ethod is not linear in the starting value u 0. Thus, it sees that our frae up to now does not cover the cg-ethod. However, there is a reedy which uses results on the cg-ethod well known in the Russian literature [7, 9]. We have to x soe notation. Let h; i be the euclidean scalar product of the nodal basis in the nite eleent space X, the induced nor will be denoted by v 2 = hv ; v i 8v 2 X : We dene the linear operator A : X! X by hav ; w i = a(v ; w ) 8v ; w 2 X ; which is represented in the nodal basis by the usual stiness atrix. The error of the cg-ethod applied to the stiness atrix can be expressed by ku? I ; u 0 k a = in p2p ; p(0)=1 kp(a )(u? u 0 )k a: Here P denotes the set of polynoials p with deg p. 7
10 The idea is now, to nd soe polynoial q ; 2 P with q ; (0) = 1, such that S ; = q ; (A ) denes a soother in the sense of (4). Since the error in energy of the cgethod is then aorized by this linear soother, the results of Section 1 are iediately valid for the cg-ethod. The choice of q ; depends on the following solution of a polynoial iniization proble. Lea 2.1. Let > 0. The Chebyshev polynoial T 2+1 has the representation T 2+1 (x) = (?1) (2 + 1) x ; (x 2 ) with a unique ; 2 P and ; (0) = 1. The polynoial ; solves the iniization proble ax p xp(x) = in! x2[0;] over all polynoials p 2 P which are noralized by p(0) = 1. The inial value is given by ax p p x ; (x) = x2[0;] : Moreover we have ax ; (x) = 1: x2[0;] A proof ay be found in the book of Shaidurov [9]. However, Shaidurov represents ; by the expression ; (x) = Y k=1 (1? x= k ); k = cos 2 ((2k? 1)=2(2 + 1)): As a fairly easy consequence Shaidurov [10] proves the following Theore 2.2. The linear operator S ; = ; (A ); = ax (A ) 8
11 satises q (i) ks ; v k a v 8v 2 X : (ii) ks ; v k a kv k a A little nite eleent theory shows that we have found a aorizing soother for the cg-ethod. Corollary 2.3. The linear operator S ; = ; (A ); = ax (A ) denes a soother in the sense of (4) with paraeter = 1. Proof. The usual inverse inequality shows that the axiu eigenvalue of the stiness atrix can be estiated by ch d?2 ; cf. [6, Theore 8.8.6]. The euclidean nor with respect to the nodal basis is related to the L 2 -nor by 1 c hd v 2 kv k 2 L 2 chd v 2 ; cf. [6, Theore 8.8.1]. Thus Theore 2.2 gives ks ; v k a c h(d?2)= v c h? kv k L 2; i.e., (4(i)) with = 1. Property (4(ii)) was already stated in Theore 2.2. With the help of this aorizing soother it is iediately clear, that Theore 1.2, Lea 1.3, Lea 1.4 and Lea 1.5 reain valid for the cascadic iteration with the cg-ethod as basic iteration, short the CCGethod of Deuhard [3]. In particular the CCG ethod is optial for d = 2; Adaptive control of the CCG-ethod. In this section we develop an adaptive control of the CCG ethod which is based on our theoretical considerations and soe additional assuptions on the faily of triangulation. 9
12 For adaptive triangulations we drop the assuption of quasi-unifority. All constants in the sequel will not depend on the quasi-unifority, but only on the shape regularity of the triangulations. In order to bound the axiu eigenvalue of the involved atrix, we copute the stiness atrix A with respect to the scaled nodal basis h (2?d)=2 ;i ;i i = 1; : : : ; n ; where f ;i g i is the usual nodal basis of X and h ;i = dia supp ;i : In this section we denote by h; i the euclidean scalar product with respect to this scaled nodal basis. Xu [11] has shown the equivalence of nors and the bound 1 X c v 2 T 2T h?2 T kv k 2 L 2 (T ) c v 2 ; h T = dia T; = ax (A ) c for the axiu eigenvalue of A. Hence, we get for the aorizing soothing iteration of the conugate gradient ethod as dened in Section 2 ks ; v k a c X h?2 T kv k 2 L 2 (T ) T 2T In order to turn this into a starting point for a theore like Theore 1.2 we ake the following two assuptions on the faily of triangulations: 1 A 1=2 : (6) (i) h?2 T ku? u?1 k 2 L 2 (T ) cku? u?1 k 2 H 1 (T ) ; 8T 2 T (ii) ku? u k c n?1=d kfk L 2: This is heuristically ustied for adaptive triangulations. The rst assuption (i) eans, that the nite eleent correction is locally of high frequency with respect to the ner triangulation. In other words, the reneent resolves changes but not ore. Thus it is a stateent of the eciency of a triangulation. Note that quasi-unifor triangulations do not accoplish assuption (ii) for probles which are not H 2 -regular. The second assuption is a stateent of optial accuracy, which is ustied by results of nonlinear approxiation theory like [8]. The sae proof as for Theore 1.2 gives the following 10
13 Lea 3.1. Assuption (6) iplies for the error of the cascadic conugate gradient iteration ku`? u `k a c 1 ku? u?1 k a c 1 n 1=d kfk L 2: We can now extend Lea 1.3 and Lea 1.4 to the adaptive case. Here we have to use additionally, that the sequence of nuber of unknowns belongs to soe kind of geoetric progression: n < 0 n n +1 1 n = 0; 1; : : : : With the choice of the iteration nubers as sallest integers for which (7)! (d+1)=2d n` ; n we get for d > 1 under assuption (6) the nal error and the coplexity ku`? u `k a c n 1=d ` n c n`: kfk L 2 However, in an adaptive algorith we do not know at level the nuber n` of nodal points at the nal level. So far our iteration is not ipleentable. But with slight changes we can still operate with it. We dene the nal level ` as the one, for which the approxiation error is below soe user given tolerance TOL. Hence assuption (6) gives us the relation ku? u k a TOL! 1=d n` ; n which leads us to replace (7) by the sallest integer with (8) ku? u k a TOL! (d+1)=2 : The actual approxiation error ku? u k a is also not known, so we replace this expression by (9) ku? u k a?1 n?1 n! 1=d 11
14 for soe estiate?1 of the previous approxiation error ku? u?1 k a, cf. [2]. This algorith is nearest to the a priori choice of the paraeters. In practice, the basic iteration can be accurate enough uch earlier than stated in theory. The crucial relation we used for the algorith was ku? u k a c ku? u?1 k a + ku?1? u?1k a ; so we turn it into a terination criterion for the basic iteration by using (8). We terinate the iteration with (10) ku? u k a! (d+1)=2 TOL + ku?1? u ku? u k k?1 a: a Here 0 < < 1 is soe safety factor. Of course we replace ku? u k a by the estiate (9) and ku?u k a, ku?1?u?1k a by soe estiate of the iteration error. Despite the fact, that our terination criterion is dierent fro the original one of Deuhard [3] we get coparable nuerical results for the CCG ethod. They share the essential feature that the iteration has to be ore accurate on coarser triangulations. However, the basis for (10) sees to be a sound cobination of theory and heuristics. Exaple. We applied the adaptive CCG ethod with TOL = 10?4 to the elliptic proble?u = 0; u?1 = 1; u?2 = n u?3 = 0 on a doain which is a unit square with slit = fx 2 R 2 : x 1 1g \ fx 2 R 2 : x 2 0:03x 1 g: The boundary pieces are? 1 = fx 2 : x 1 = 1; x 2 0:03g;? 2 = fx 2 : x 1 = 1; x 2?0:03g; and? 3 n (? 1 [? 2 ). Figure 1 shows a typical triangulation and isolines of the solution. In Figure 2 we have drawn the nuber of iterations which were used for the corresponding nuber of unknowns n. We copared three dierent ipleentations: CCG1 : the CCG ethod with terination criterion (10). 12
15 Fig. 1. Typical adaptive triangulation with isolines of solution CCG2 : the CCG ethod with Deuhard's terination criterion [3]. CSGS : an adaptive cascadic iteration using syetric Gauss-Seidel as basic iteration. We ipleented the terination criterion (10) in this case also. We observe that the CCG1 and CCG2 ipleentations are coparable, the CCG1 needs one adaptive step less in order to achieve the tolerance TOL. The perforance of the CSGS ipleentation is a little bit soother and gives copletely satisfactory results. In fact, if one looks at the real CPU-tie needed for the achieved accuracy ku? u k a, one gets nearly indistinguishable curves for all three ipleentations! Acknowledgeent. The author thanks Rudi Beck for propt coputational assistence. REFERENCES [1] R. E. Bank and T. F. Dupont. An optial order process for solving elliptic nite eleent equations. Math. Cop. 36, 967{975 (1981). [2] F. A. Borneann, B. Erdann, and R. Kornhuber. A posteriori error estiates for elliptic probles in two and three space diensions. SIAM J. Nuer. Anal. (subitted). [3] P. Deuflhard. Cascadic conugate gradient ethods for elliptic partial dierential equations. Algorith and nuerical results. In D. Keyes and J. Xu, editors, 13
16 20 15 CCG 1 CCG 2 CSGS iterations nuber of nodes Fig. 2. Nuber of iterations vs. nuber of unknowns \Proceedings of the 7th International Conference on Doain Decoposition Methods 1993". AMS, Providence (1994). [4] P. Deuflhard, P. Leinen, and H. Yserentant. Concepts of an adaptive hierarchical nite eleent code. IMPACT Copt. Sci. Engrg. 1, 3{35 (1989). [5] W. Hackbusch. \Multi-Grid Methods and Applications". Springer-Verlag, Berlin, Heidelberg, New York (1985). [6] W. Hackbusch. \Theory and Nuerical Treatent of Elliptic Dierential Equations". Springer-Verlag, Berlin, Heidelberg, New York (1992). [7] V. P. Il'in. Soe estiates for conugate gradient ethods. USSR Coput. Math. and Math. Phys. 16, 22{30 (1976). [8] P. Oswald. On function spaces related to nite eleent approxiation theory. Z. Anal. Anwend. 9, 43{64 (1990). [9] V. V. Shaidurov. \Multigrid Methods for Finite Eleents". Kluwer Acadeic Publishers, Dordrecht, Boston, London (1994). [10] V. V. Shaidurov. \Soe estiates of the rate of convergence for the cascadic conugate-gradient ethod". Otto-von-Guericke-Universitat, Magdeburg (1994). Preprint. [11] J. Xu. \Theory of Multilevel Methods". Departent of Matheatics, Pennsylvania State University, University Park (1989). Report No. AM
ON THE TWO-LEVEL PRECONDITIONING IN LEAST SQUARES METHOD
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Matheatical Sciences 04,, p. 7 5 ON THE TWO-LEVEL PRECONDITIONING IN LEAST SQUARES METHOD M a t h e a t i c s Yu. A. HAKOPIAN, R. Z. HOVHANNISYAN
More informationGeneralized AOR Method for Solving System of Linear Equations. Davod Khojasteh Salkuyeh. Department of Mathematics, University of Mohaghegh Ardabili,
Australian Journal of Basic and Applied Sciences, 5(3): 35-358, 20 ISSN 99-878 Generalized AOR Method for Solving Syste of Linear Equations Davod Khojasteh Salkuyeh Departent of Matheatics, University
More informationModel Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
More informatione-companion ONLY AVAILABLE IN ELECTRONIC FORM
OPERATIONS RESEARCH doi 10.1287/opre.1070.0427ec pp. ec1 ec5 e-copanion ONLY AVAILABLE IN ELECTRONIC FORM infors 07 INFORMS Electronic Copanion A Learning Approach for Interactive Marketing to a Custoer
More informationANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION
ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION HAO GAO AND HONGKAI ZHAO Abstract. We analyze a nuerical algorith for solving radiative transport equation with vacuu or reflection boundary
More informationOn the Communication Complexity of Lipschitzian Optimization for the Coordinated Model of Computation
journal of coplexity 6, 459473 (2000) doi:0.006jco.2000.0544, available online at http:www.idealibrary.co on On the Counication Coplexity of Lipschitzian Optiization for the Coordinated Model of Coputation
More informationA new type of lower bound for the largest eigenvalue of a symmetric matrix
Linear Algebra and its Applications 47 7 9 9 www.elsevier.co/locate/laa A new type of lower bound for the largest eigenvalue of a syetric atrix Piet Van Mieghe Delft University of Technology, P.O. Box
More informationA Bernstein-Markov Theorem for Normed Spaces
A Bernstein-Markov Theore for Nored Spaces Lawrence A. Harris Departent of Matheatics, University of Kentucky Lexington, Kentucky 40506-0027 Abstract Let X and Y be real nored linear spaces and let φ :
More informationNew upper bound for the B-spline basis condition number II. K. Scherer. Institut fur Angewandte Mathematik, Universitat Bonn, Bonn, Germany.
New upper bound for the B-spline basis condition nuber II. A proof of de Boor's 2 -conjecture K. Scherer Institut fur Angewandte Matheati, Universitat Bonn, 535 Bonn, Gerany and A. Yu. Shadrin Coputing
More informationCh 12: Variations on Backpropagation
Ch 2: Variations on Backpropagation The basic backpropagation algorith is too slow for ost practical applications. It ay take days or weeks of coputer tie. We deonstrate why the backpropagation algorith
More information. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe
PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal
More informationUniform Approximation and Bernstein Polynomials with Coefficients in the Unit Interval
Unifor Approxiation and Bernstein Polynoials with Coefficients in the Unit Interval Weiang Qian and Marc D. Riedel Electrical and Coputer Engineering, University of Minnesota 200 Union St. S.E. Minneapolis,
More informationwhich together show that the Lax-Milgram lemma can be applied. (c) We have the basic Galerkin orthogonality
UPPSALA UNIVERSITY Departent of Inforation Technology Division of Scientific Coputing Solutions to exa in Finite eleent ethods II 14-6-5 Stefan Engblo, Daniel Elfverson Question 1 Note: a inus sign in
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationA BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM FOR A NONLINEAR SINGULARLY PERTURBED PARABOLIC PROBLEM
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volue 3, Nuber 2, Pages 211 231 c 2006 Institute for Scientific Coputing and Inforation A BLOCK MONOTONE DOMAIN DECOMPOSITION ALGORITHM FOR A NONLINEAR
More informationM ath. Res. Lett. 15 (2008), no. 2, c International Press 2008 SUM-PRODUCT ESTIMATES VIA DIRECTED EXPANDERS. Van H. Vu. 1.
M ath. Res. Lett. 15 (2008), no. 2, 375 388 c International Press 2008 SUM-PRODUCT ESTIMATES VIA DIRECTED EXPANDERS Van H. Vu Abstract. Let F q be a finite field of order q and P be a polynoial in F q[x
More informationMax-Product Shepard Approximation Operators
Max-Product Shepard Approxiation Operators Barnabás Bede 1, Hajie Nobuhara 2, János Fodor 3, Kaoru Hirota 2 1 Departent of Mechanical and Syste Engineering, Bánki Donát Faculty of Mechanical Engineering,
More informationList Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
More informationANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS FOR MILDLY STRUCTURED GRIDS
MATHEMATICS OF COMPUTATION Volue 73, Nuber 247, Pages 1139 1152 S 0025-5718(03)01600-4 Article electronically published on August 19, 2003 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS FOR MILDLY
More informationRECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS MEMBRANE
Proceedings of ICIPE rd International Conference on Inverse Probles in Engineering: Theory and Practice June -8, 999, Port Ludlow, Washington, USA : RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS
More informationPolygonal Designs: Existence and Construction
Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G
More informationApproximation by Piecewise Constants on Convex Partitions
Aroxiation by Piecewise Constants on Convex Partitions Oleg Davydov Noveber 4, 2011 Abstract We show that the saturation order of iecewise constant aroxiation in L nor on convex artitions with N cells
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationBipartite subgraphs and the smallest eigenvalue
Bipartite subgraphs and the sallest eigenvalue Noga Alon Benny Sudaov Abstract Two results dealing with the relation between the sallest eigenvalue of a graph and its bipartite subgraphs are obtained.
More informationA Better Algorithm For an Ancient Scheduling Problem. David R. Karger Steven J. Phillips Eric Torng. Department of Computer Science
A Better Algorith For an Ancient Scheduling Proble David R. Karger Steven J. Phillips Eric Torng Departent of Coputer Science Stanford University Stanford, CA 9435-4 Abstract One of the oldest and siplest
More informationLeast Squares Fitting of Data
Least Squares Fitting of Data David Eberly, Geoetric Tools, Redond WA 98052 https://www.geoetrictools.co/ This work is licensed under the Creative Coons Attribution 4.0 International License. To view a
More informationExplicit solution of the polynomial least-squares approximation problem on Chebyshev extrema nodes
Explicit solution of the polynoial least-squares approxiation proble on Chebyshev extrea nodes Alfredo Eisinberg, Giuseppe Fedele Dipartiento di Elettronica Inforatica e Sisteistica, Università degli Studi
More informationTHE POLYNOMIAL REPRESENTATION OF THE TYPE A n 1 RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n
THE POLYNOMIAL REPRESENTATION OF THE TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n SHEELA DEVADAS AND YI SUN Abstract. We study the polynoial representation of the rational Cherednik algebra
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationAlgorithms for parallel processor scheduling with distinct due windows and unit-time jobs
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 57, No. 3, 2009 Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs A. JANIAK 1, W.A. JANIAK 2, and
More informationPerturbation on Polynomials
Perturbation on Polynoials Isaila Diouf 1, Babacar Diakhaté 1 & Abdoul O Watt 2 1 Départeent Maths-Infos, Université Cheikh Anta Diop, Dakar, Senegal Journal of Matheatics Research; Vol 5, No 3; 2013 ISSN
More informationDistributed Subgradient Methods for Multi-agent Optimization
1 Distributed Subgradient Methods for Multi-agent Optiization Angelia Nedić and Asuan Ozdaglar October 29, 2007 Abstract We study a distributed coputation odel for optiizing a su of convex objective functions
More informationarxiv: v1 [math.na] 10 Oct 2016
GREEDY GAUSS-NEWTON ALGORITHM FOR FINDING SPARSE SOLUTIONS TO NONLINEAR UNDERDETERMINED SYSTEMS OF EQUATIONS MÅRTEN GULLIKSSON AND ANNA OLEYNIK arxiv:6.395v [ath.na] Oct 26 Abstract. We consider the proble
More informationA RESTARTED KRYLOV SUBSPACE METHOD FOR THE EVALUATION OF MATRIX FUNCTIONS. 1. Introduction. The evaluation of. f(a)b, where A C n n, b C n (1.
A RESTARTED KRYLOV SUBSPACE METHOD FOR THE EVALUATION OF MATRIX FUNCTIONS MICHAEL EIERMANN AND OLIVER G. ERNST Abstract. We show how the Arnoldi algorith for approxiating a function of a atrix ties a vector
More informationOn Poset Merging. 1 Introduction. Peter Chen Guoli Ding Steve Seiden. Keywords: Merging, Partial Order, Lower Bounds. AMS Classification: 68W40
On Poset Merging Peter Chen Guoli Ding Steve Seiden Abstract We consider the follow poset erging proble: Let X and Y be two subsets of a partially ordered set S. Given coplete inforation about the ordering
More informationThis model assumes that the probability of a gap has size i is proportional to 1/i. i.e., i log m e. j=1. E[gap size] = i P r(i) = N f t.
CS 493: Algoriths for Massive Data Sets Feb 2, 2002 Local Models, Bloo Filter Scribe: Qin Lv Local Models In global odels, every inverted file entry is copressed with the sae odel. This work wells when
More informationIntelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes
More informationSupport Vector Machine Classification of Uncertain and Imbalanced data using Robust Optimization
Recent Researches in Coputer Science Support Vector Machine Classification of Uncertain and Ibalanced data using Robust Optiization RAGHAV PAT, THEODORE B. TRAFALIS, KASH BARKER School of Industrial Engineering
More informationComputational and Statistical Learning Theory
Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher
More informationFPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
Matheatical Prograing anuscript No. (will be inserted by the editor) Jesús A. De Loera Rayond Heecke Matthias Köppe Robert Weisantel FPTAS for optiizing polynoials over the ixed-integer points of polytopes
More informationIterative Linear Solvers and Jacobian-free Newton-Krylov Methods
Eric de Sturler Iterative Linear Solvers and Jacobian-free Newton-Krylov Methods Eric de Sturler Departent of Matheatics, Virginia Tech www.ath.vt.edu/people/sturler/index.htl sturler@vt.edu Efficient
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationTopic 5a Introduction to Curve Fitting & Linear Regression
/7/08 Course Instructor Dr. Rayond C. Rup Oice: A 337 Phone: (95) 747 6958 E ail: rcrup@utep.edu opic 5a Introduction to Curve Fitting & Linear Regression EE 4386/530 Coputational ethods in EE Outline
More informationRESTARTED FULL ORTHOGONALIZATION METHOD FOR SHIFTED LINEAR SYSTEMS
BIT Nuerical Matheatics 43: 459 466, 2003. 2003 Kluwer Acadeic Publishers. Printed in The Netherlands 459 RESTARTED FULL ORTHOGONALIZATION METHOD FOR SHIFTED LINEAR SYSTEMS V. SIMONCINI Dipartiento di
More informationNecessity of low effective dimension
Necessity of low effective diension Art B. Owen Stanford University October 2002, Orig: July 2002 Abstract Practitioners have long noticed that quasi-monte Carlo ethods work very well on functions that
More informationSharp Time Data Tradeoffs for Linear Inverse Problems
Sharp Tie Data Tradeoffs for Linear Inverse Probles Saet Oyak Benjain Recht Mahdi Soltanolkotabi January 016 Abstract In this paper we characterize sharp tie-data tradeoffs for optiization probles used
More informationA Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness
A Note on Scheduling Tall/Sall Multiprocessor Tasks with Unit Processing Tie to Miniize Maxiu Tardiness Philippe Baptiste and Baruch Schieber IBM T.J. Watson Research Center P.O. Box 218, Yorktown Heights,
More informationCOS 424: Interacting with Data. Written Exercises
COS 424: Interacting with Data Hoework #4 Spring 2007 Regression Due: Wednesday, April 18 Written Exercises See the course website for iportant inforation about collaboration and late policies, as well
More informationTight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography
Tight Bounds for axial Identifiability of Failure Nodes in Boolean Network Toography Nicola Galesi Sapienza Università di Roa nicola.galesi@uniroa1.it Fariba Ranjbar Sapienza Università di Roa fariba.ranjbar@uniroa1.it
More informationADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE
ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE CHRISTOPHER J. HILLAR Abstract. A long-standing conjecture asserts that the polynoial p(t = Tr(A + tb ] has nonnegative coefficients whenever is
More informationOn the Use of A Priori Information for Sparse Signal Approximations
ITS TECHNICAL REPORT NO. 3/4 On the Use of A Priori Inforation for Sparse Signal Approxiations Oscar Divorra Escoda, Lorenzo Granai and Pierre Vandergheynst Signal Processing Institute ITS) Ecole Polytechnique
More informationLecture 21. Interior Point Methods Setup and Algorithm
Lecture 21 Interior Point Methods In 1984, Kararkar introduced a new weakly polynoial tie algorith for solving LPs [Kar84a], [Kar84b]. His algorith was theoretically faster than the ellipsoid ethod and
More informationFixed-to-Variable Length Distribution Matching
Fixed-to-Variable Length Distribution Matching Rana Ali Ajad and Georg Böcherer Institute for Counications Engineering Technische Universität München, Gerany Eail: raa2463@gail.co,georg.boecherer@tu.de
More informationGeneralized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.
Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, 2006 1 Introduction There is a well developed theory (see [5,
More informationAn Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes
An Optial Faily of Exponentially Accurate One-Bit Siga-Delta Quantization Schees Percy Deift C. Sinan Güntürk Felix Kraher January 2, 200 Abstract Siga-Delta odulation is a popular ethod for analog-to-digital
More informationAlgebraic Montgomery-Yang problem: the log del Pezzo surface case
c 2014 The Matheatical Society of Japan J. Math. Soc. Japan Vol. 66, No. 4 (2014) pp. 1073 1089 doi: 10.2969/jsj/06641073 Algebraic Montgoery-Yang proble: the log del Pezzo surface case By DongSeon Hwang
More informationANALYSIS OF HALL-EFFECT THRUSTERS AND ION ENGINES FOR EARTH-TO-MOON TRANSFER
IEPC 003-0034 ANALYSIS OF HALL-EFFECT THRUSTERS AND ION ENGINES FOR EARTH-TO-MOON TRANSFER A. Bober, M. Guelan Asher Space Research Institute, Technion-Israel Institute of Technology, 3000 Haifa, Israel
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationNumerically repeated support splitting and merging phenomena in a porous media equation with strong absorption. Kenji Tomoeda
Journal of Math-for-Industry, Vol. 3 (C-), pp. Nuerically repeated support splitting and erging phenoena in a porous edia equation with strong absorption To the eory of y friend Professor Nakaki. Kenji
More informationNumerical issues in the implementation of high order polynomial multidomain penalty spectral Galerkin methods for hyperbolic conservation laws
Nuerical issues in the ipleentation of high order polynoial ultidoain penalty spectral Galerkin ethods for hyperbolic conservation laws Sigal Gottlieb 1 and Jae-Hun Jung 1, 1 Departent of Matheatics, University
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationThe Transactional Nature of Quantum Information
The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.
More information1. INTRODUCTION AND RESULTS
SOME IDENTITIES INVOLVING THE FIBONACCI NUMBERS AND LUCAS NUMBERS Wenpeng Zhang Research Center for Basic Science, Xi an Jiaotong University Xi an Shaanxi, People s Republic of China (Subitted August 00
More informationTail estimates for norms of sums of log-concave random vectors
Tail estiates for nors of sus of log-concave rando vectors Rados law Adaczak Rafa l Lata la Alexander E. Litvak Alain Pajor Nicole Toczak-Jaegerann Abstract We establish new tail estiates for order statistics
More informationTR DUAL-PRIMAL FETI METHODS FOR LINEAR ELASTICITY. September 14, 2004
TR2004-855 DUAL-PRIMAL FETI METHODS FOR LINEAR ELASTICITY AXEL KLAWONN AND OLOF B WIDLUND Septeber 14, 2004 Abstract Dual-Prial FETI ethods are nonoverlapping doain decoposition ethods where soe of the
More informationAbout the definition of parameters and regimes of active two-port networks with variable loads on the basis of projective geometry
About the definition of paraeters and regies of active two-port networks with variable loads on the basis of projective geoetry PENN ALEXANDR nstitute of Electronic Engineering and Nanotechnologies "D
More informationtime time δ jobs jobs
Approxiating Total Flow Tie on Parallel Machines Stefano Leonardi Danny Raz y Abstract We consider the proble of optiizing the total ow tie of a strea of jobs that are released over tie in a ultiprocessor
More informationarxiv: v1 [cs.ds] 3 Feb 2014
arxiv:40.043v [cs.ds] 3 Feb 04 A Bound on the Expected Optiality of Rando Feasible Solutions to Cobinatorial Optiization Probles Evan A. Sultani The Johns Hopins University APL evan@sultani.co http://www.sultani.co/
More informationA Note on Online Scheduling for Jobs with Arbitrary Release Times
A Note on Online Scheduling for Jobs with Arbitrary Release Ties Jihuan Ding, and Guochuan Zhang College of Operations Research and Manageent Science, Qufu Noral University, Rizhao 7686, China dingjihuan@hotail.co
More informationFast Montgomery-like Square Root Computation over GF(2 m ) for All Trinomials
Fast Montgoery-like Square Root Coputation over GF( ) for All Trinoials Yin Li a, Yu Zhang a, a Departent of Coputer Science and Technology, Xinyang Noral University, Henan, P.R.China Abstract This letter
More informationShannon Sampling II. Connections to Learning Theory
Shannon Sapling II Connections to Learning heory Steve Sale oyota echnological Institute at Chicago 147 East 60th Street, Chicago, IL 60637, USA E-ail: sale@athberkeleyedu Ding-Xuan Zhou Departent of Matheatics,
More informationAnalyzing Simulation Results
Analyzing Siulation Results Dr. John Mellor-Cruey Departent of Coputer Science Rice University johnc@cs.rice.edu COMP 528 Lecture 20 31 March 2005 Topics for Today Model verification Model validation Transient
More informationTheore A. Let n (n 4) be arcoplete inial iersed hypersurface in R n+. 4(n ) Then there exists a constant C (n) = (n 2)n S (n) > 0 such that if kak n d
Gap theores for inial subanifolds in R n+ Lei Ni Departent of atheatics Purdue University West Lafayette, IN 47907 99 atheatics Subject Classification 53C2 53C42 Introduction Let be a copact iersed inial
More informationStochastic Subgradient Methods
Stochastic Subgradient Methods Lingjie Weng Yutian Chen Bren School of Inforation and Coputer Science University of California, Irvine {wengl, yutianc}@ics.uci.edu Abstract Stochastic subgradient ethods
More informationHomework 3 Solutions CSE 101 Summer 2017
Hoework 3 Solutions CSE 0 Suer 207. Scheduling algoriths The following n = 2 jobs with given processing ties have to be scheduled on = 3 parallel and identical processors with the objective of iniizing
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More informationNORMAL MATRIX POLYNOMIALS WITH NONSINGULAR LEADING COEFFICIENTS
NORMAL MATRIX POLYNOMIALS WITH NONSINGULAR LEADING COEFFICIENTS NIKOLAOS PAPATHANASIOU AND PANAYIOTIS PSARRAKOS Abstract. In this paper, we introduce the notions of weakly noral and noral atrix polynoials,
More informationConvex Programming for Scheduling Unrelated Parallel Machines
Convex Prograing for Scheduling Unrelated Parallel Machines Yossi Azar Air Epstein Abstract We consider the classical proble of scheduling parallel unrelated achines. Each job is to be processed by exactly
More informationANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR THE PHASE FIELD MODEL AND APPROXIMATION OF ITS SHARP INTERFACE LIMITS
ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR THE PHASE FIELD MODEL AND APPROXIMATION OF ITS SHARP INTERFACE LIMITS XIAOBING FENG AND ANDREAS PROHL Abstract. We propose and analyze a fully discrete
More informationarxiv: v1 [math.nt] 14 Sep 2014
ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row
More informationThe Hilbert Schmidt version of the commutator theorem for zero trace matrices
The Hilbert Schidt version of the coutator theore for zero trace atrices Oer Angel Gideon Schechtan March 205 Abstract Let A be a coplex atrix with zero trace. Then there are atrices B and C such that
More informationThe Simplex Method is Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate
The Siplex Method is Strongly Polynoial for the Markov Decision Proble with a Fixed Discount Rate Yinyu Ye April 20, 2010 Abstract In this note we prove that the classic siplex ethod with the ost-negativereduced-cost
More informationDedicated to Richard S. Varga on the occasion of his 80th birthday.
MATRICES, MOMENTS, AND RATIONAL QUADRATURE G. LÓPEZ LAGOMASINO, L. REICHEL, AND L. WUNDERLICH Dedicated to Richard S. Varga on the occasion of his 80th birthday. Abstract. Many probles in science and engineering
More informationResearch Article On the Isolated Vertices and Connectivity in Random Intersection Graphs
International Cobinatorics Volue 2011, Article ID 872703, 9 pages doi:10.1155/2011/872703 Research Article On the Isolated Vertices and Connectivity in Rando Intersection Graphs Yilun Shang Institute for
More informationUniversal algorithms for learning theory Part II : piecewise polynomial functions
Universal algoriths for learning theory Part II : piecewise polynoial functions Peter Binev, Albert Cohen, Wolfgang Dahen, and Ronald DeVore Deceber 6, 2005 Abstract This paper is concerned with estiating
More informationPrerequisites. We recall: Theorem 2 A subset of a countably innite set is countable.
Prerequisites 1 Set Theory We recall the basic facts about countable and uncountable sets, union and intersection of sets and iages and preiages of functions. 1.1 Countable and uncountable sets We can
More informationExtension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels
Extension of CSRSM for the Paraetric Study of the Face Stability of Pressurized Tunnels Guilhe Mollon 1, Daniel Dias 2, and Abdul-Haid Soubra 3, M.ASCE 1 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique
More informationUCSD Spring School lecture notes: Continuous-time quantum computing
UCSD Spring School lecture notes: Continuous-tie quantu coputing David Gosset 1 Efficient siulation of quantu dynaics Quantu echanics is described atheatically using linear algebra, so at soe level is
More informationC na (1) a=l. c = CO + Clm + CZ TWO-STAGE SAMPLE DESIGN WITH SMALL CLUSTERS. 1. Introduction
TWO-STGE SMPLE DESIGN WITH SMLL CLUSTERS Robert G. Clark and David G. Steel School of Matheatics and pplied Statistics, University of Wollongong, NSW 5 ustralia. (robert.clark@abs.gov.au) Key Words: saple
More informationBernoulli Wavelet Based Numerical Method for Solving Fredholm Integral Equations of the Second Kind
ISSN 746-7659, England, UK Journal of Inforation and Coputing Science Vol., No., 6, pp.-9 Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind S. C. Shiralashetti*,
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationEMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS
EMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS Jochen Till, Sebastian Engell, Sebastian Panek, and Olaf Stursberg Process Control Lab (CT-AST), University of Dortund,
More informationHybrid System Identification: An SDP Approach
49th IEEE Conference on Decision and Control Deceber 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Hybrid Syste Identification: An SDP Approach C Feng, C M Lagoa, N Ozay and M Sznaier Abstract The
More informationTHE CONSTRUCTION OF GOOD EXTENSIBLE RANK-1 LATTICES. 1. Introduction We are interested in approximating a high dimensional integral [0,1]
MATHEMATICS OF COMPUTATION Volue 00, Nuber 0, Pages 000 000 S 0025-578(XX)0000-0 THE CONSTRUCTION OF GOOD EXTENSIBLE RANK- LATTICES JOSEF DICK, FRIEDRICH PILLICHSHAMMER, AND BENJAMIN J. WATERHOUSE Abstract.
More informationA note on the multiplication of sparse matrices
Cent. Eur. J. Cop. Sci. 41) 2014 1-11 DOI: 10.2478/s13537-014-0201-x Central European Journal of Coputer Science A note on the ultiplication of sparse atrices Research Article Keivan Borna 12, Sohrab Aboozarkhani
More informationOn the Bell- Kochen -Specker paradox
On the Bell- Kochen -Specker paradox Koji Nagata and Tadao Nakaura Departent of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea E-ail: ko_i_na@yahoo.co.jp Departent of Inforation
More informationA MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY
A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,
More informationCharacterization of the Line Complexity of Cellular Automata Generated by Polynomial Transition Rules. Bertrand Stone
Characterization of the Line Coplexity of Cellular Autoata Generated by Polynoial Transition Rules Bertrand Stone Abstract Cellular autoata are discrete dynaical systes which consist of changing patterns
More informationPattern Recognition and Machine Learning. Learning and Evaluation for Pattern Recognition
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lesson 1 4 October 2017 Outline Learning and Evaluation for Pattern Recognition Notation...2 1. The Pattern Recognition
More information